On weighted complex Randers metrics

In this paper we introduce the weighted complex Randers metric F=h+\sum_{i=1}^{m}\lvert B_{i}\rvert^{1/i} on a complex manifold M, here h is a Hermitian metric on M and B_{i}, i=1,\ldots, m are holomorphic symmetric forms of weights i on M, respectively. These metrics are special case of jet metric...

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Bibliographic Details
Published in:Osaka journal of mathematics 2011-09, Vol.48 (no. 3), p.589-612
Main Authors: Wong, Pit-Mann, Zhong, Chunping
Format: Article
Language:English
Subjects:
32Q10
53C56
53C60
Online Access:Get full text
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Summary:In this paper we introduce the weighted complex Randers metric F=h+\sum_{i=1}^{m}\lvert B_{i}\rvert^{1/i} on a complex manifold M, here h is a Hermitian metric on M and B_{i}, i=1,\ldots, m are holomorphic symmetric forms of weights i on M, respectively. These metrics are special case of jet metric studied in Chandler--Wong [6]. Our main theorem is that the holomorphic sectional curvature \mathrm{hbsc}_{F} of F is always less or equal to \mathrm{hbsc}_{h}. Using this result we obtain a rigidity result, that is, a compact complex manifold M of complex dimension n with a weighted complex Randers metric F of positive constant holomorphic sectional curvature is isomorphic to \mathbb{P}^{n}.